Efficient algorithms for the Riemann-Roch problem and for addition in the Jacobian of a curve
Journal of Symbolic Computation
Regular Article: The Diameter of Sparse Random Graphs
Advances in Applied Mathematics
Introduction to Algorithms
Computing in the jacobian of a plane algebraic curve
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
Linear algebra algorithms for divisors on an algebraic curve
Mathematics of Computation
Algebraic curves and cryptography
Finite Fields and Their Applications
Computing L-polynomials of non-hyperelliptic genus 4 and 5 curves (abstract only)
ACM Communications in Computer Algebra
An L (1/3 + ε) Algorithm for the Discrete Logarithm Problem for Low Degree Curves
EUROCRYPT '07 Proceedings of the 26th annual international conference on Advances in Cryptology
Journal of Symbolic Computation
Isogenies and the discrete logarithm problem in Jacobians of genus 3 hyperelliptic curves
EUROCRYPT'08 Proceedings of the theory and applications of cryptographic techniques 27th annual international conference on Advances in cryptology
Correspondences on hyperelliptic curves and applications to the discrete logarithm
SIIS'11 Proceedings of the 2011 international conference on Security and Intelligent Information Systems
Group law computations on jacobians of hyperelliptic curves
SAC'11 Proceedings of the 18th international conference on Selected Areas in Cryptography
EUROCRYPT'12 Proceedings of the 31st Annual international conference on Theory and Applications of Cryptographic Techniques
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We present an index calculus algorithm which is particularly well suited to solve the discrete logarithm problem (DLP) in degree 0 class groups of curves over finite fields which are represented by plane models of small degree. A heuristic analysis of our algorithm indicates that asymptotically for varying q, “almost all” instances of the DLP in degree 0 class groups of curves represented by plane models of a fixed degree d ≥4 over $\mathbb{F}_{q}$ can be solved in an expected time of $\tilde{O}(q^{2-2/(d-2)})$. Additionally we provide a method to represent “sufficiently general” (non-hyperelliptic) curves of genus g ≥3 by plane models of degree g+1. We conclude that on heuristic grounds, “almost all” instances of the DLP in degree 0 class groups of (non-hyperelliptic) curves of a fixed genus g ≥3 (represented initially by plane models of bounded degree) can be solved in an expected time of $\tilde{O}(q^{2 -2/(g-1)})$.